Question: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{-6n^2 - 72n - 120}{-6n^2 + 12n + 48}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {-6(n^2 + 12n + 20)} {-6(n^2 - 2n - 8)} $ $ x = \dfrac{6}{6} \cdot \dfrac{n^2 + 12n + 20}{n^2 - 2n - 8} $ Simplify: $ x = \dfrac{n^2 + 12n + 20}{n^2 - 2n - 8}$ Next factor the numerator and denominator. $ x = \dfrac{(n + 2)(n + 10)}{(n + 2)(n - 4)}$ Assuming $n \neq -2$ , we can cancel the $n + 2$ $ x = \dfrac{n + 10}{n - 4}$ Therefore: $ x = \dfrac{ n + 10 }{ n - 4 }$, $n \neq -2$